Two digit numbers that yields remainder $1$ when divided by $4$ forms sequence
$13,17,21,........97$
This sequence is an A.P with first term $a=13$ common difference $d=4$ and
$t_n=97$
$\Rightarrow\:a+(n-1)d=97$
$\Rightarrow\:13+(n-1)4=97$
$\Rightarrow\:n-1=\large\frac{84}{4}$$=21$
or $n=22$
$\therefore$ Sum of the required numbers =$13+17+21+.......97$
$=\large\frac{n}{2}$$(t_n+a)=11(97+13)=11\times 110=1210$